The Prime Geodesic Theorem and Bounds for Character Sums

Abstract

We establish the prime geodesic theorem for the modular surface with exponent $\frac{2}{3}+\varepsilon$, improving upon the long-standing exponent $\frac{25}{36}+\varepsilon$ of Soundararajan-Young (2013). This was previously known conditionally on the generalised Lindel\"{o}f hypothesis for quadratic Dirichlet $L$-functions. Our argument goes through a well-trodden trail via the automorphic machinery, and refines the techniques of Iwaniec (1984) and Cai (2002) to a maximum extent. A key ingredient is an asymptotic for bilinear forms with a counting function in Kloosterman sums via hybrid Weyl-strength subconvex bounds for quadratic Dirichlet $L$-functions due to Young (2017), zero density estimates due to Heath-Brown (1995) near the edge of the critical strip, and an asymptotic for averages of Zagier $L$-series due to Balkanova-Frolenkov-Risager (2022). Furthermore, we strengthen our exponent to $\frac{5}{8}+\varepsilon$ conditionally on the generalised Lindel\"{o}f hypothesis for quadratic Dirichlet $L$-functions, which breaks the existing barrier.